Also, if the strings (or columns) of $A_{n \times n}$ do not form the basis in ${\mathbb R}^n$, it means that some of the strings / columns are linearly dependent, and thus $\mathrm{rk}(A) &lt n$, $|A| = 0$, so that A is not invertible.

From the other hand, if the strings (or columns) of $A_{n \times n}$ do form the basis, it means that they are linearly independent, and $|A| \ne 0$. In that case, you can find $A^{-1}$ by the Cramer's rule.